Fock space

We will now construct a vector space, which we will informally call Fock space4 and denote it by \(\mathcal{F}(M)\). One suitable set of basis vectors for this vector space are the occupation number vectors (ONVs) \(\require{physics} \ket{\vb{k}}\): \[\begin{equation} \ket{\vb{k}} % = \ket{k_1, k_2, \ldots, k_M} % \qquad \qquad % k_P = 0, 1 % \thinspace , \end{equation}\] in which \(k_P\) is called the occupation number of the \(P\)-th spinor. Given an occupation number vector \(\ket{\vb{k}}\), it is possible to calculate the number of electrons \(N\) as the sum of all the occupation numbers: \[\begin{equation} N = \sum_{P=1}^M k_P \thinspace . \end{equation}\] Every occupation number vector \(\ket{\vb{k}}\) has a one-to-one correspondence to a Slater determinant (cfr. section \(\ref{sec:states}\)), in the sense that \(k_P = 1\) if spinor \(\phi_P\) is present in the Slater determinant, and \(k_P = 0\) if it is not. We now equip our newly constructed vector space with an inner product that is consistent with the overlap between single Slater determinants: \[\begin{align} \braket{\vb{k}}{\vb{m}} % &= \prod_{P=1}^M \delta_{k_P m_P} \label{eq:inner_product_fock_space_orthonormal} \\ % &= \delta_{\vb{k}, \vb{m}} % \thinspace , \end{align}\] such that the resolution of the identity for the orthonormal basis vectors \(\qty{\ket{\vb{k}}}\) can be written as \[\begin{equation} 1 = \sum_{\vb{k}} \dyad{\vb{k}} \thinspace , \end{equation}\] where the sum is over all ONVs for all numbers of electrons. The (antisymmetric part of the) total Fock space \(\mathcal{F}(M)\) can be written as a direct sum of subspaces \(\mathcal{F}(M, N)\): \[\begin{equation} \mathcal{F}(M) % = \bigoplus_{N=0}^M \mathcal{F}(M, N) % \thinspace , \end{equation}\] which are spanned by \[\begin{equation} \label{eq:dim_fock_M_N} \dim \mathcal{F}(M,N) = \binom{M}{N} \end{equation}\] basis vectors \(\ket{\vb{k}}\), being equal to the the number of ways to distribute \(N\) electrons over \(M\) spinors (with exclusion due to the Pauli exclusion principle). We can say that \(\mathcal{F}(M,N)\) represents the antisymmetrized \(N\)-electron sector of the \(M\)-orbital Fock space. The subspace \(\mathcal{F}(M,0)\) is thus spanned by a special ONV: \[\begin{equation} \ket{0_1, 0_2, \ldots, 0_M} % = \ket{\text{vac}} % \thinspace , \end{equation}\] which contains no electrons and which we will call the physical vacuum state. By the definition of the inner product of Fock space (cfr. equation \(\eqref{eq:inner_product_fock_space_orthonormal}\)), the vacuum state is normalized to unity.

Up until now, we haven’t introduced any approximations at all if we let \(M \rightarrow \infty\). On the contrary, by using a finite \(M\)-dimensional single particle basis, the nature of the approximations used in quantum chemistry becomes clear: the electronic wave function is only expanded inside a smaller Fock subspace.

  1. and recognize that \(\mathcal{F}(M)\) only describes the odd or antisymmetric part of the real Fock space↩︎